2 GRAPHICAL ANALYSIS — SUMMARY
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2.1
Graphical Analysis — summary
Cobweb diagrams
Cobweb diagrams allow us to iterate a function by entirely graphical means and without
having to resort to analytic or numerical methods.
Consider the function f (x) plotted below. Starting from a point x0 , we can find the
next iterate of the function, x1 = f (x0 ), simply by drawing a vertical line to the plot of the
function. x1 can be then marked on the vertical axis by drawing a horizontal line from the
point of intersection.
(x0 , x1 )
x1
x0
In order to find x2 = f (x1 ), we need to move the point x1 marked on the vertical axis to the
same point on the horizontal axis. We do this by finding the intersection of the horizontal
line with the line y = x. Since the horizontal line has equation y = x1 , this intersection will
occur at the point (x1 , x1 ). Drawing a vertical line down to the horizontal axis will then
mark the point x1 .
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Dynamical Systems and Chaos — 620341
y=x
(x0 , x1 )
x1
x1
x0
Given that we now have x1 on the horizontal axis we can find the point x2 = f (x1 ) by
drawing a vertical line up to the plot of the function.
y=x
x2
(x1 , x2 )
(x0 , x1 )
x1
x0
This proceedure can then be iterated to generate a “cobweb diagram” which shows the
positions of future iterates of the function.
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2 GRAPHICAL ANALYSIS — SUMMARY
In this example we see that the iterates settle into a 2-cycle (which is marked in blue and
red).
Below is a cobweb diagram of the iterates of x0 = 0.3 under cos(x):
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Dynamical Systems and Chaos — 620341
1.2
1
0.8
0.6
0.4
0.2
0
–0.2
0.2
0.6
0.4
0.8
1
1.2
x
–0.2
We see that the iterates converge to the fixed point at x ≈ 0.739 . . .
Below is a cobweb diagram of the iterates of x0 = 0.5 under x2 − 1.1:
1
0.5
–1
–0.5
0.5
1
x
–0.5
–1
We see that the iterates converge to the two cycle at {p+ , p− } ≈ {0.0916, −1.0916}.
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2 GRAPHICAL ANALYSIS — SUMMARY
2.2
Phase portraits
In the picture below we show the iterates of f (x) = x3 . This function has three fixed points,
x = −1, 0, 1. If |x0 | > 1 then iterates diverge to infinity, while if |x0 | < 1 then the iterates
converge to the fixed point at x = 0.
2
1
–1
–0.5
0.5
1
x
–1
We can summarise the information about these orbits on the real line by the following
diagram:
−1
0
+1
This is a “phase portrait”. It shows how points on the real line move to new points on the
real line under application of f (x). While this diagram gives no more information than the
cobweb diagram, we are able to use phase portraits for higher dimensional systems where
we are unable to apply cobweb diagrams.
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